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<!DOCTYPE html>
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<title>Scalar Magnitudes</title>
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<div class="container">
<h1>Scalar Magnitudes</h1>
<blockquote>
Scalar Magnitudes are invariant scalar quantities that conserve their value and form under transformations of translation and rotation, or changes between coordinate systems (Cartesian, polar, spherical, etc.).
</blockquote>
<hr>
<h2>I. Definitions</h2>
<h3>0. Vectorial Magnitudes</h3>
<p>The vectorial position ($\vec{r}_{ij}$), vectorial velocity ($\vec{v}_{ij}$), and vectorial acceleration ($\vec{a}_{ij}$) of two particles $i$ and $j$ are given by:</p>
<table>
<thead>
<tr>
<th>Vectorial Magnitude</th>
<th>Definition</th>
<th>Derivation</th>
</tr>
</thead>
<tbody>
<tr>
<td>Position ($\vec{r}_{ij}$)</td>
<td>$\vec{r}_{ij} \doteq (\vec{r}_i - \vec{r}_j)$</td>
<td>(Fundamental definition)</td>
</tr>
<tr>
<td>Velocity ($\vec{v}_{ij}$)</td>
<td>$\vec{v}_{ij} \doteq (\vec{v}_i - \vec{v}_j)$</td>
<td>$\vec{v}_{ij} \doteq \frac{d(\vec{r}_{ij})}{dt}$</td>
</tr>
<tr>
<td>Acceleration ($\vec{a}_{ij}$)</td>
<td>$\vec{a}_{ij} \doteq (\vec{a}_i - \vec{a}_j)$</td>
<td>$\vec{a}_{ij} \doteq \frac{d^2(\vec{r}_{ij})}{dt^2}$</td>
</tr>
</tbody>
</table>
<h3>1. Scalar Magnitudes</h3>
<p>The scalar position ($\tau_{ij}$), scalar velocity ($\dot{\tau}_{ij}$), and scalar acceleration ($\ddot{\tau}_{ij}$) of two particles $i$ and $j$ are given by:</p>
<table>
<thead>
<tr>
<th>Scalar Magnitude</th>
<th>Definition</th>
<th>Derivation</th>
</tr>
</thead>
<tbody>
<tr>
<td>Position ($\tau_{ij}$)</td>
<td>$\tau_{ij} \doteq \frac{1}{2} \vec{r}_{ij} \cdot \vec{r}_{ij}$</td>
<td>(Fundamental definition)</td>
</tr>
<tr>
<td>Velocity ($\dot{\tau}_{ij}$)</td>
<td>$\dot{\tau}_{ij} \doteq \vec{v}_{ij} \cdot \vec{r}_{ij}$</td>
<td>$\dot{\tau}_{ij} \doteq \frac{d(\tau_{ij})}{dt}$</td>
</tr>
<tr>
<td>Acceleration ($\ddot{\tau}_{ij}$)</td>
<td>$\ddot{\tau}_{ij} \doteq \vec{a}_{ij} \cdot \vec{r}_{ij} + \vec{v}_{ij} \cdot \vec{v}_{ij}$</td>
<td>$\ddot{\tau}_{ij} \doteq \frac{d^2(\tau_{ij})}{dt^2}$</td>
</tr>
</tbody>
</table>
<hr>
<h2>II. Scalar Invariance Demonstrations</h2>
<h3>0. Vectorial Transformations (Absolute)</h3>
<p>The vectorial position ($\vec{r}'_i$), vectorial velocity ($\vec{v}'_i$), and vectorial acceleration ($\vec{a}'_i$) of a particle $i$ with respect to a Reference Frame $S'$, whose origin $O'$ is at the vectorial position $\vec{r}_{O'}$ with respect to another Reference Frame $S$, are given by:</p>
<div class="math-block">
$\vec{r}'_i = \vec{r}_i - \vec{r}_{O'}$
</div>
<div class="math-block">
$\vec{v}'_i = \vec{v}_i - \vec{v}_{O'} - \vec{\omega} \times (\vec{r}_i - \vec{r}_{O'})$
</div>
<div class="math-block">
$\vec{a}'_i = \vec{a}_i - \vec{a}_{O'} - 2 \vec{\omega} \times (\vec{v}_i - \vec{v}_{O'}) + \vec{\omega} \times (\vec{\omega} \times (\vec{r}_i - \vec{r}_{O'})) - \vec{\alpha} \times (\vec{r}_i - \vec{r}_{O'}) $
</div>
<p>Where $\vec{r}_i$, $\vec{v}_i$, and $\vec{a}_i$ are the vectorial position, velocity, and acceleration of particle $i$ with respect to Frame $S$; and $\vec{\omega}$ and $\vec{\alpha}$ are the angular velocity and angular acceleration of Frame $S'$ with respect to Frame $S$.</p>
<p><strong>Note</strong></p><p>If $\vec{m}'_i = \vec{n}_i$ then:</p><p>$\dfrac{d(\vec{m}'_i)}{dt} = \dfrac{d(\vec{n}_i)}{dt} - \vec{\omega} \times \vec{n}_i$</p>
<hr>
<h3>1. Scalar Position Invariance ($\tau_{ij}$)</h3>
<p>The Scalar Position $\tau_{ij}$ is invariant under rotation and translation because the magnitude of the relative vector is preserved.</p>
<div class="math-block">
$\tau_{ij} = \frac{1}{2} (\vec{r}_i - \vec{r}_j) \cdot (\vec{r}_i - \vec{r}_j)$
</div>
<div class="math-block">
$\tau'_{ij} = \frac{1}{2} (\vec{r}'_i - \vec{r}'_j) \cdot (\vec{r}'_i - \vec{r}'_j)$
</div>
<div class="math-block">
$\text{Since } (\vec{r}_i - \vec{r}_j) = (\vec{r}'_i - \vec{r}'_j)$
</div>
<div class="math-block">
$\text{Because } \vec{r}'_i = \vec{r}_i - \vec{r}_{O'} \text{ and } \vec{r}'_j = \vec{r}_j - \vec{r}_{O'} \text{ (The relative position vector is independent of the Frame's origin.)}$
</div>
<div class="math-block">
$\tau'_{ij} = \frac{1}{2} (\vec{r}_i - \vec{r}_j) \cdot (\vec{r}_i - \vec{r}_j)$
</div>
<div class="math-block">
$\therefore \tau'_{ij} = \tau_{ij}$
</div>
<hr>
<h3>2. Scalar Velocity Invariance ($\dot{\tau}_{ij}$)</h3>
<p>The Scalar Velocity $\dot{\tau}_{ij}$ is invariant because the cross-product generated by the angular velocity ($\vec{\omega}$) is perpendicular to the relative position vector, resulting in a zero scalar product.</p>
<div class="math-block">
$\dot{\tau}_{ij} = (\vec{v}_i - \vec{v}_j) \cdot (\vec{r}_i - \vec{r}_j)$
</div>
<div class="math-block">
$\dot{\tau}'_{ij} = (\vec{v}'_i - \vec{v}'_j) \cdot (\vec{r}'_i - \vec{r}'_j)$
</div>
<div class="math-block">
$\dot{\tau}'_{ij} = \left( (\vec{v}_i - \vec{v}_j) - \vec{\omega} \times (\vec{r}_i - \vec{r}_j) \right) \cdot (\vec{r}_i - \vec{r}_j)$
</div>
<div class="math-block">
$\text{Since } (-\ \vec{\omega} \times (\vec{r}_i - \vec{r}_j)) \cdot (\vec{r}_i - \vec{r}_j) = 0 \text{ (The rotational term is orthogonal to the relative position vector.)}$
</div>
<div class="math-block">
$\text{Because } (\vec{A} \times \vec{B}) \cdot \vec{B} = 0 \text{ (Property of the Scalar Triple Product)}$
</div>
<div class="math-block">
$\dot{\tau}'_{ij} = (\vec{v}_i - \vec{v}_j) \cdot (\vec{r}_i - \vec{r}_j)$
</div>
<div class="math-block">
$\therefore \dot{\tau}'_{ij} = \dot{\tau}_{ij}$
</div>
<hr>
<h3>3. Scalar Acceleration Invariance ($\ddot{\tau}_{ij}$)</h3>
<p>The Scalar Acceleration $\ddot{\tau}_{ij}$ is invariant because all inertial terms (Angular Acceleration, Coriolis, and Centrifugal) mutually cancel due to the properties of the vector and scalar products.</p>
<div class="math-block">
$\ddot{\tau}_{ij} = (\vec{a}_i - \vec{a}_j) \cdot (\vec{r}_i - \vec{r}_j) + (\vec{v}_i - \vec{v}_j) \cdot (\vec{v}_i - \vec{v}_j)$
</div>
<div class="math-block">
$\ddot{\tau}'_{ij} = (\vec{a}'_i - \vec{a}'_j) \cdot (\vec{r}'_i - \vec{r}'_j) + (\vec{v}'_i - \vec{v}'_j) \cdot (\vec{v}'_i - \vec{v}'_j)$
</div>
<div class="math-block">
$\ddot{\tau}'_{ij} = \left[ (\vec{a}_i - \vec{a}_j) - 2 \vec{\omega} \times (\vec{v}_i - \vec{v}_j) + \vec{\omega} \times (\vec{\omega} \times (\vec{r}_i - \vec{r}_j)) - \vec{\alpha} \times (\vec{r}_i - \vec{r}_j) \right] \cdot (\vec{r}_i - \vec{r}_j) + \left[ (\vec{v}_i - \vec{v}_j) - \vec{\omega} \times (\vec{r}_i - \vec{r}_j) \right] \cdot \left[ (\vec{v}_i - \vec{v}_j) - \vec{\omega} \times (\vec{r}_i - \vec{r}_j) \right]$
</div>
<div class="math-block">
$\text{Since } - (\vec{\alpha} \times (\vec{r}_i - \vec{r}_j)) \cdot (\vec{r}_i - \vec{r}_j) = 0 \text{ (Angular acceleration term cancels)}$
</div>
<div class="math-block">
$\text{Because } (\vec{A} \times \vec{B}) \cdot \vec{B} = 0 \text{ (Property of the Scalar Triple Product)}$
</div>
<div class="math-block">
$\text{Since } - 2 (\vec{\omega} \times (\vec{v}_i - \vec{v}_j)) \cdot (\vec{r}_i - \vec{r}_j) - 2 (\vec{v}_i - \vec{v}_j) \cdot (\vec{\omega} \times (\vec{r}_i - \vec{r}_j)) = 0 \text{ (Coriolis terms cancel)}$
</div>
<div class="math-block">
$\text{Because } (\vec{A} \times \vec{B}) \cdot \vec{C} = \vec{A} \cdot (\vec{B} \times \vec{C}) \text{ (Cyclic Permutation Property of the Scalar Triple Product)}$
</div>
<div class="math-block">
$\text{Since } + (\vec{\omega} \times (\vec{\omega} \times (\vec{r}_i - \vec{r}_j))) \cdot (\vec{r}_i - \vec{r}_j) + (\vec{\omega} \times (\vec{r}_i - \vec{r}_j)) \cdot (\vec{\omega} \times (\vec{r}_i - \vec{r}_j)) = 0 \text{ (Centrifugal terms cancel)}$
</div>
<div class="math-block">
$\text{Since } + \vec{P} \cdot (\vec{r}_i - \vec{r}_j) + E = 0$
</div>
<div class="math-block">
$\text{Because } \vec{P} = \vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C}) \ \vec{B} - (\vec{A} \cdot \vec{B}) \ \vec{C} \text{ (Vector Triple Product)}$
</div>
<div class="math-block">
$\text{Because } E = (\vec{A} \times \vec{B}) \cdot (\vec{A} \times \vec{B}) = (\vec{A} \cdot \vec{A}) \ (\vec{B} \cdot \vec{B}) - (\vec{A} \cdot \vec{B})^2 \text{ (Lagrange's Identity)}$
</div>
<div class="math-block">
$\ddot{\tau}'_{ij} = (\vec{a}_i - \vec{a}_j) \cdot (\vec{r}_i - \vec{r}_j) + (\vec{v}_i - \vec{v}_j) \cdot (\vec{v}_i - \vec{v}_j)$
</div>
<div class="math-block">
$\therefore \ddot{\tau}'_{ij} = \ddot{\tau}_{ij}$
</div>
<hr>
<h2>III. Fundamental Relations</h2>
<h3>1. Radial Magnitudes Relations</h3>
<p>The scalar magnitudes ($\tau_{ij}, \dot{\tau}_{ij}, \ddot{\tau}_{ij}$) expressed using radial magnitudes ($r_{ij}$) are given by:</p>
<table>
<thead>
<tr>
<th>Scalar Magnitude</th>
<th>Relation with Radial Magnitudes</th>
</tr>
</thead>
<tbody>
<tr>
<td>$\tau_{ij}$</td>
<td>$\tau_{ij} = \frac{1}{2} r_{ij}^2$</td>
</tr>
<tr>
<td>$\dot{\tau}_{ij}$</td>
<td>$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$</td>
</tr>
<tr>
<td>$\ddot{\tau}_{ij}$</td>
<td>$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$</td>
</tr>
</tbody>
</table>
<h3>2. Polar Magnitudes Relations</h3>
<p>The scalar magnitudes ($\tau_{ij}, \dot{\tau}_{ij}, \ddot{\tau}_{ij}$) expressed using polar magnitudes ($r_{ij}$) are given by:</p>
<table>
<thead>
<tr>
<th>Scalar Magnitude</th>
<th>Relation with Polar Magnitudes</th>
</tr>
</thead>
<tbody>
<tr>
<td>$\tau_{ij}$</td>
<td>$\tau_{ij} = \frac{1}{2} r_{ij}^2$</td>
</tr>
<tr>
<td>$\dot{\tau}_{ij}$</td>
<td>$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$</td>
</tr>
<tr>
<td>$\ddot{\tau}_{ij}$</td>
<td>$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$</td>
</tr>
</tbody>
</table>
<h3>3. Cylindrical Magnitudes Relations</h3>
<p>The scalar magnitudes ($\tau_{ij}, \dot{\tau}_{ij}, \ddot{\tau}_{ij}$) expressed using cylindrical magnitudes ($r_{ij}$) are given by:</p>
<table>
<thead>
<tr>
<th>Scalar Magnitude</th>
<th>Relation with Cylindrical Magnitudes</th>
</tr>
</thead>
<tbody>
<tr>
<td>$\tau_{ij}$</td>
<td>$\tau_{ij} = \frac{1}{2} r_{ij}^2$</td>
</tr>
<tr>
<td>$\dot{\tau}_{ij}$</td>
<td>$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$</td>
</tr>
<tr>
<td>$\ddot{\tau}_{ij}$</td>
<td>$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$</td>
</tr>
</tbody>
</table>
<h3>4. Circular Magnitudes Relations</h3>
<p>The scalar magnitudes ($\tau_{ij}, \dot{\tau}_{ij}, \ddot{\tau}_{ij}$) expressed using circular magnitudes ($r_{ij}$) are given by:</p>
<table>
<thead>
<tr>
<th>Scalar Magnitude</th>
<th>Relation with Circular Magnitudes</th>
</tr>
</thead>
<tbody>
<tr>
<td>$\tau_{ij}$</td>
<td>$\tau_{ij} = \frac{1}{2} r_{ij}^2$</td>
</tr>
<tr>
<td>$\dot{\tau}_{ij}$</td>
<td>$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$</td>
</tr>
<tr>
<td>$\ddot{\tau}_{ij}$</td>
<td>$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$</td>
</tr>
</tbody>
</table>
<h3>5. Spherical Magnitudes Relations</h3>
<p>The scalar magnitudes ($\tau_{ij}, \dot{\tau}_{ij}, \ddot{\tau}_{ij}$) expressed using spherical magnitudes ($r_{ij}$) are given by:</p>
<table>
<thead>
<tr>
<th>Scalar Magnitude</th>
<th>Relation with Spherical Magnitudes</th>
</tr>
</thead>
<tbody>
<tr>
<td>$\tau_{ij}$</td>
<td>$\tau_{ij} = \frac{1}{2} r_{ij}^2$</td>
</tr>
<tr>
<td>$\dot{\tau}_{ij}$</td>
<td>$\dot{\tau}_{ij} = r_{ij} \dot{r}_{ij}$</td>
</tr>
<tr>
<td>$\ddot{\tau}_{ij}$</td>
<td>$\ddot{\tau}_{ij} = r_{ij} \ddot{r}_{ij} + \dot{r}_{ij}^2$</td>
</tr>
</tbody>
</table>
<hr>
<h2>IV. Bibliography</h2>
<ol>
<li>A. Torassa, <em>A Group of Invariant Equations</em> (2014). <a href="https://atorassa.github.io/physics-authors/torassa/english/pdf/34.pdf">[PDF]</a></li>
<li>A. Torassa, <em>A Reformulation of Classical Mechanics</em> (2014). <a href="https://atorassa.github.io/physics-authors/torassa/english/pdf/45.pdf">[PDF]</a></li>
<li>A. Tobla, <em>Linear, Radial & Scalar Magnitudes</em> (2015). <a href="https://atorassa.github.io/physics-authors/tobla/english/pdf/01.pdf">[PDF]</a></li>
<li>A. Tobla, <em>A Reformulation of Classical Mechanics</em> (2024). <a href="https://atorassa.github.io/physics-authors/tobla/english/pdf/02.pdf">[PDF]</a></li>
</ol>
</div>
</body>
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